The Critique of Pure Reason; Part 7

The Critique of Pure Reason; Part 7 Lyrics

CHAPTER II. System of all Principles of the Pure Understanding.

In the foregoing chapter we have merely considered the general
conditions under which alone the transcendental faculty of judgement
is justified in using the pure conceptions of the understanding for
synthetical judgements. Our duty at present is to exhibit in systematic
connection those judgements which the understanding really produces a
priori. For this purpose, our table of the categories will certainly
afford us the natural and safe guidance. For it is precisely the
categories whose application to possible experience must constitute all
pure a priori cognition of the understanding; and the relation of which
to sensibility will, on that very account, present us with a complete
and systematic catalogue of all the transcendental principles of the use
of the understanding.

Principles a priori are so called, not merely because they contain
in themselves the grounds of other judgements, but also because they
themselves are not grounded in higher and more general cognitions. This
peculiarity, however, does not raise them altogether above the need of
a proof. For although there could be found no higher cognition, and
therefore no objective proof, and although such a principle rather
serves as the foundation for all cognition of the object, this by no
means hinders us from drawing a proof from the subjective sources of the
possibility of the cognition of an object. Such a proof is necessary,
moreover, because without it the principle might be liable to the
imputation of being a mere gratuitous assertion.

In the second place, we shall limit our investigations to those
principles which relate to the categories. For as to the principles
of transcendental aesthetic, according to which space and time are
the conditions of the possibility of things as phenomena, as also the
restriction of these principles, namely, that they cannot be applied to
objects as things in themselves--these, of course, do not fall within
the scope of our present inquiry. In like manner, the principles of
mathematical science form no part of this system, because they are
all drawn from intuition, and not from the pure conception of the
understanding. The possibility of these principles, however, will
necessarily be considered here, inasmuch as they are synthetical
judgements a priori, not indeed for the purpose of proving their
accuracy and apodeictic certainty, which is unnecessary, but merely to
render conceivable and deduce the possibility of such evident a priori
cognitions.

But we shall have also to speak of the principle of analytical
judgements, in opposition to synthetical judgements, which is the proper
subject of our inquiries, because this very opposition will free the
theory of the latter from all ambiguity, and place it clearly before our
eyes in its true nature.

SYSTEM OF THE PRINCIPLES OF THE PURE UNDERSTANDING.

SECTION I. Of the Supreme Principle of all Analytical Judgements.

Whatever may be the content of our cognition, and in whatever manner
our cognition may be related to its object, the universal, although only
negative conditions of all our judgements is that they do not contradict
themselves; otherwise these judgements are in themselves (even without
respect to the object) nothing. But although there may exist no
contradiction in our judgement, it may nevertheless connect conceptions
in such a manner that they do not correspond to the object, or without
any grounds either a priori or a posteriori for arriving at such a
judgement, and thus, without being self-contradictory, a judgement may
nevertheless be either false or groundless.

Now, the proposition: "No subject can have a predicate that contradicts
it," is called the principle of contradiction, and is a universal but
purely negative criterion of all truth. But it belongs to logic alone,
because it is valid of cognitions, merely as cognitions and without
respect to their content, and declares that the contradiction entirely
nullifies them. We can also, however, make a positive use of this
principle, that is, not merely to banish falsehood and error (in so far
as it rests upon contradiction), but also for the cognition of truth.
For if the judgement is analytical, be it affirmative or negative,
its truth must always be recognizable by means of the principle of
contradiction. For the contrary of that which lies and is cogitated
as conception in the cognition of the object will be always properly
negatived, but the conception itself must always be affirmed of the
object, inasmuch as the contrary thereof would be in contradiction to
the object.

We must therefore hold the principle of contradiction to be the
universal and fully sufficient Principle of all analytical cognition.
But as a sufficient criterion of truth, it has no further utility or
authority. For the fact that no cognition can be at variance with this
principle without nullifying itself, constitutes this principle the sine
qua non, but not the determining ground of the truth of our cognition.
As our business at present is properly with the synthetical part of our
knowledge only, we shall always be on our guard not to transgress this
inviolable principle; but at the same time not to expect from it any
direct assistance in the establishment of the truth of any synthetical
proposition.

There exists, however, a formula of this celebrated principle--a
principle merely formal and entirely without content--which contains a
synthesis that has been inadvertently and quite unnecessarily mixed up
with it. It is this: "It is impossible for a thing to be and not to be
at the same time." Not to mention the superfluousness of the addition of
the word impossible to indicate the apodeictic certainty, which ought to
be self-evident from the proposition itself, the proposition is affected
by the condition of time, and as it were says: "A thing = A, which is
something = B, cannot at the same time be non-B." But both, B as well
as non-B, may quite well exist in succession. For example, a man who is
young cannot at the same time be old; but the same man can very well
be at one time young, and at another not young, that is, old. Now the
principle of contradiction as a merely logical proposition must not
by any means limit its application merely to relations of time, and
consequently a formula like the preceding is quite foreign to its
true purpose. The misunderstanding arises in this way. We first of all
separate a predicate of a thing from the conception of the thing, and
afterwards connect with this predicate its opposite, and hence do
not establish any contradiction with the subject, but only with its
predicate, which has been conjoined with the subject synthetically--a
contradiction, moreover, which obtains only when the first and second
predicate are affirmed in the same time. If I say: "A man who is
ignorant is not learned," the condition "at the same time" must be
added, for he who is at one time ignorant, may at another be learned.
But if I say: "No ignorant man is a learned man," the proposition is
analytical, because the characteristic ignorance is now a constituent
part of the conception of the subject; and in this case the
negative proposition is evident immediately from the proposition of
contradiction, without the necessity of adding the condition "the
same time." This is the reason why I have altered the formula of this
principle--an alteration which shows very clearly the nature of an
analytical proposition.

SECTION II. Of the Supreme Principle of all Synthetical Judgements.

The explanation of the possibility of synthetical judgements is a task
with which general logic has nothing to do; indeed she needs not even
be acquainted with its name. But in transcendental logic it is the most
important matter to be dealt with--indeed the only one, if the question
is of the possibility of synthetical judgements a priori, the conditions
and extent of their validity. For when this question is fully decided,
it can reach its aim with perfect ease, the determination, to wit, of
the extent and limits of the pure understanding.

In an analytical judgement I do not go beyond the given conception,
in order to arrive at some decision respecting it. If the judgement is
affirmative, I predicate of the conception only that which was already
cogitated in it; if negative, I merely exclude from the conception its
contrary. But in synthetical judgements, I must go beyond the given
conception, in order to cogitate, in relation with it, something quite
different from that which was cogitated in it, a relation which is
consequently never one either of identity or contradiction, and by means
of which the truth or error of the judgement cannot be discerned merely
from the judgement itself.

Granted, then, that we must go out beyond a given conception, in order
to compare it synthetically with another, a third thing is necessary, in
which alone the synthesis of two conceptions can originate. Now what
is this tertium quid that is to be the medium of all synthetical
judgements? It is only a complex in which all our representations are
contained, the internal sense to wit, and its form a priori, time.

The synthesis of our representations rests upon the imagination; their
synthetical unity (which is requisite to a judgement), upon the unity
of apperception. In this, therefore, is to be sought the possibility of
synthetical judgements, and as all three contain the sources of a priori
representations, the possibility of pure synthetical judgements also;
nay, they are necessary upon these grounds, if we are to possess
a knowledge of objects, which rests solely upon the synthesis of
representations.

If a cognition is to have objective reality, that is, to relate to an
object, and possess sense and meaning in respect to it, it is necessary
that the object be given in some way or another. Without this, our
conceptions are empty, and we may indeed have thought by means of them,
but by such thinking we have not, in fact, cognized anything, we have
merely played with representation. To give an object, if this expression
be understood in the sense of "to present" the object, not mediately
but immediately in intuition, means nothing else than to apply the
representation of it to experience, be that experience real or only
possible. Space and time themselves, pure as these conceptions are from
all that is empirical, and certain as it is that they are represented
fully a priori in the mind, would be completely without objective
validity, and without sense and significance, if their necessary use
in the objects of experience were not shown. Nay, the representation
of them is a mere schema, that always relates to the reproductive
imagination, which calls up the objects of experience, without which
they have no meaning. And so it is with all conceptions without
distinction.

The possibility of experience is, then, that which gives objective
reality to all our a priori cognitions. Now experience depends upon the
synthetical unity of phenomena, that is, upon a synthesis according to
conceptions of the object of phenomena in general, a synthesis without
which experience never could become knowledge, but would be merely a
rhapsody of perceptions, never fitting together into any connected text,
according to rules of a thoroughly united (possible) consciousness, and
therefore never subjected to the transcendental and necessary unity
of apperception. Experience has therefore for a foundation, a priori
principles of its form, that is to say, general rules of unity in
the synthesis of phenomena, the objective reality of which rules, as
necessary conditions even of the possibility of experience can
which rules, as necessary conditions--even of the possibility of
experience--can always be shown in experience. But apart from this
relation, a priori synthetical propositions are absolutely impossible,
because they have no third term, that is, no pure object, in which the
synthetical unity can exhibit the objective reality of its conceptions.

Although, then, respecting space, or the forms which productive
imagination describes therein, we do cognize much a priori in
synthetical judgements, and are really in no need of experience for this
purpose, such knowledge would nevertheless amount to nothing but a busy
trifling with a mere chimera, were not space to be considered as the
condition of the phenomena which constitute the material of external
experience. Hence those pure synthetical judgements do relate, though
but mediately, to possible experience, or rather to the possibility of
experience, and upon that alone is founded the objective validity of
their synthesis.

While then, on the one hand, experience, as empirical synthesis, is
the only possible mode of cognition which gives reality to all other
synthesis; on the other hand, this latter synthesis, as cognition a
priori, possesses truth, that is, accordance with its object, only in
so far as it contains nothing more than what is necessary to the
synthetical unity of experience.

Accordingly, the supreme principle of all synthetical judgements is:
"Every object is subject to the necessary conditions of the synthetical
unity of the manifold of intuition in a possible experience."

A priori synthetical judgements are possible when we apply the formal
conditions of the a priori intuition, the synthesis of the imagination,
and the necessary unity of that synthesis in a transcendental
apperception, to a possible cognition of experience, and say: "The
conditions of the possibility of experience in general are at the same
time conditions of the possibility of the objects of experience, and
have, for that reason, objective validity in an a priori synthetical
judgement."

SECTION III. Systematic Representation of all Synthetical Principles of
the Pure Understanding.

That principles exist at all is to be ascribed solely to the pure
understanding, which is not only the faculty of rules in regard to that
which happens, but is even the source of principles according to which
everything that can be presented to us as an object is necessarily
subject to rules, because without such rules we never could attain
to cognition of an object. Even the laws of nature, if they are
contemplated as principles of the empirical use of the understanding,
possess also a characteristic of necessity, and we may therefore at
least expect them to be determined upon grounds which are valid a
priori and antecedent to all experience. But all laws of nature, without
distinction, are subject to higher principles of the understanding,
inasmuch as the former are merely applications of the latter to
particular cases of experience. These higher principles alone therefore
give the conception, which contains the necessary condition, and, as it
were, the exponent of a rule; experience, on the other hand, gives the
case which comes under the rule.

There is no danger of our mistaking merely empirical principles for
principles of the pure understanding, or conversely; for the character
of necessity, according to conceptions which distinguish the latter,
and the absence of this in every empirical proposition, how extensively
valid soever it may be, is a perfect safeguard against confounding
them. There are, however, pure principles a priori, which nevertheless I
should not ascribe to the pure understanding--for this reason, that they
are not derived from pure conceptions, but (although by the mediation
of the understanding) from pure intuitions. But understanding is the
faculty of conceptions. Such principles mathematical science possesses,
but their application to experience, consequently their objective
validity, nay the possibility of such a priori synthetical cognitions
(the deduction thereof) rests entirely upon the pure understanding.

On this account, I shall not reckon among my principles those of
mathematics; though I shall include those upon the possibility and
objective validity a priori, of principles of the mathematical science,
which, consequently, are to be looked upon as the principle of these,
and which proceed from conceptions to intuition, and not from intuition
to conceptions.

In the application of the pure conceptions of the understanding to
possible experience, the employment of their synthesis is either
mathematical or dynamical, for it is directed partly on the intuition
alone, partly on the existence of a phenomenon. But the a priori
conditions of intuition are in relation to a possible experience
absolutely necessary, those of the existence of objects of a possible
empirical intuition are in themselves contingent. Hence the principles
of the mathematical use of the categories will possess a character of
absolute necessity, that is, will be apodeictic; those, on the other
hand, of the dynamical use, the character of an a priori necessity
indeed, but only under the condition of empirical thought in an
experience, therefore only mediately and indirectly. Consequently
they will not possess that immediate evidence which is peculiar to the
former, although their application to experience does not, for that
reason, lose its truth and certitude. But of this point we shall be
better able to judge at the conclusion of this system of principles.

The table of the categories is naturally our guide to the table of
principles, because these are nothing else than rules for the objective
employment of the former. Accordingly, all principles of the pure
understanding are:

1
Axioms
of Intuition

2
Anticipations
of Perception

3
Analogies
of Experience

4
Postulates of
Empirical Thought
in general

These appellations I have chosen advisedly, in order that we might
not lose sight of the distinctions in respect of the evidence and the
employment of these principles. It will, however, soon appear that--a
fact which concerns both the evidence of these principles, and the
a priori determination of phenomena--according to the categories of
quantity and quality (if we attend merely to the form of these), the
principles of these categories are distinguishable from those of the two
others, in as much as the former are possessed of an intuitive, but
the latter of a merely discursive, though in both instances a complete,
certitude. I shall therefore call the former mathematical, and the
latter dynamical principles.* It must be observed, however, that by
these terms I mean just as little in the one case the principles of
mathematics as those of general (physical) dynamics in the other. I have
here in view merely the principles of the pure understanding, in
their application to the internal sense (without distinction of the
representations given therein), by means of which the sciences of
mathematics and dynamics become possible. Accordingly, I have named
these principles rather with reference to their application than their
content; and I shall now proceed to consider them in the order in which
they stand in the table.

1. AXIOMS OF INTUITION.

The principle of these is: All Intuitions are Extensive Quantities.

PROOF.

All phenomena contain, as regards their form, an intuition in space and
time, which lies a priori at the foundation of all without exception.
Phenomena, therefore, cannot be apprehended, that is, received into
empirical consciousness otherwise than through the synthesis of a
manifold, through which the representations of a determinate space
or time are generated; that is to say, through the composition of the
homogeneous and the consciousness of the synthetical unity of this
manifold (homogeneous). Now the consciousness of a homogeneous manifold
in intuition, in so far as thereby the representation of an object
is rendered possible, is the conception of a quantity (quanti).
Consequently, even the perception of an object as phenomenon is possible
only through the same synthetical unity of the manifold of the given
sensuous intuition, through which the unity of the composition of the
homogeneous manifold in the conception of a quantity is cogitated;
that is to say, all phenomena are quantities, and extensive quantities,
because as intuitions in space or time they must be represented by
means of the same synthesis through which space and time themselves are
determined.

An extensive quantity I call that wherein the representation of the
parts renders possible (and therefore necessarily antecedes) the
representation of the whole. I cannot represent to myself any line,
however small, without drawing it in thought, that is, without
generating from a point all its parts one after another, and in this
way alone producing this intuition. Precisely the same is the case with
every, even the smallest, portion of time. I cogitate therein only the
successive progress from one moment to another, and hence, by means of
the different portions of time and the addition of them, a determinate
quantity of time is produced. As the pure intuition in all phenomena
is either time or space, so is every phenomenon in its character of
intuition an extensive quantity, inasmuch as it can only be cognized
in our apprehension by successive synthesis (from part to part). All
phenomena are, accordingly, to be considered as aggregates, that is, as
a collection of previously given parts; which is not the case with
every sort of quantities, but only with those which are represented and
apprehended by us as extensive.

On this successive synthesis of the productive imagination, in the
generation of figures, is founded the mathematics of extension, or
geometry, with its axioms, which express the conditions of sensuous
intuition a priori, under which alone the schema of a pure conception of
external intuition can exist; for example, "be tween two points only one
straight line is possible," "two straight lines cannot enclose a space,"
etc. These are the axioms which properly relate only to quantities
(quanta) as such.

But, as regards the quantity of a thing (quantitas), that is to say, the
answer to the question: "How large is this or that object?" although, in
respect to this question, we have various propositions synthetical and
immediately certain (indemonstrabilia); we have, in the proper sense of
the term, no axioms. For example, the propositions: "If equals be added
to equals, the wholes are equal"; "If equals be taken from equals,
the remainders are equal"; are analytical, because I am immediately
conscious of the identity of the production of the one quantity with
the production of the other; whereas axioms must be a priori synthetical
propositions. On the other hand, the self-evident propositions as to the
relation of numbers, are certainly synthetical but not universal, like
those of geometry, and for this reason cannot be called axioms, but
numerical formulae. That 7 + 5 = 12 is not an analytical proposition.
For neither in the representation of seven, nor of five, nor of the
composition of the two numbers, do I cogitate the number twelve.
(Whether I cogitate the number in the addition of both, is not at
present the question; for in the case of an analytical proposition,
the only point is whether I really cogitate the predicate in the
representation of the subject.) But although the proposition is
synthetical, it is nevertheless only a singular proposition. In so far
as regard is here had merely to the synthesis of the homogeneous (the
units), it cannot take place except in one manner, although our use
of these numbers is afterwards general. If I say: "A triangle can
be constructed with three lines, any two of which taken together are
greater than the third," I exercise merely the pure function of the
productive imagination, which may draw the lines longer or shorter and
construct the angles at its pleasure. On the contrary, the number seven
is possible only in one manner, and so is likewise the number twelve,
which results from the synthesis of seven and five. Such propositions,
then, cannot be termed axioms (for in that case we should have an
infinity of these), but numerical formulae.

This transcendental principle of the mathematics of phenomena greatly
enlarges our a priori cognition. For it is by this principle alone that
pure mathematics is rendered applicable in all its precision to objects
of experience, and without it the validity of this application would not
be so self-evident; on the contrary, contradictions and confusions have
often arisen on this very point. Phenomena are not things in themselves.
Empirical intuition is possible only through pure intuition (of space
and time); consequently, what geometry affirms of the latter, is
indisputably valid of the former. All evasions, such as the statement
that objects of sense do not conform to the rules of construction in
space (for example, to the rule of the infinite divisibility of lines or
angles), must fall to the ground. For, if these objections hold good, we
deny to space, and with it to all mathematics, objective validity, and
no longer know wherefore, and how far, mathematics can be applied to
phenomena. The synthesis of spaces and times as the essential form of
all intuition, is that which renders possible the apprehension of a
phenomenon, and therefore every external experience, consequently all
cognition of the objects of experience; and whatever mathematics in its
pure use proves of the former, must necessarily hold good of the latter.
All objections are but the chicaneries of an ill-instructed reason,
which erroneously thinks to liberate the objects of sense from the
formal conditions of our sensibility, and represents these, although
mere phenomena, as things in themselves, presented as such to our
understanding. But in this case, no a priori synthetical cognition of
them could be possible, consequently not through pure conceptions of
space and the science which determines these conceptions, that is to
say, geometry, would itself be impossible.

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